Timeline for Intuitive and/or philosophical explanation for set theory paradoxes
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2011 at 14:59 | comment | added | Andreas Blass | Platonists use temporal metaphors, but I doubt that they view sets as owing their existence (creation, construction, etc.) to the process described by such a metaphor. The sets are there anyway, and we (may) organize them into a hierarchy. | |
| Mar 11, 2011 at 22:00 | comment | added | Flash Sheridan | I seem to have run into what I’ll call Flash’s Paradox, of the set of all non-self-membered sets equinumerous to the universe; I can hope that a similar extension of Church’s original theory would avoid the difficulty. | |
| Mar 11, 2011 at 21:59 | comment | added | Flash Sheridan | Admittedly the standard formalism does possess impressively unreasonable effectiveness, and the various programs of “Fixing Frege” (e.g., also as a basis for arithmetic) are still well short of the dominant paradigm. My own attempt to extend Church’s theory (with the singleton function as a set), though it’s equiconsistent with ZF as far as it goes, seems unlikely to be able to go far enough for the needs of category theory (e.g., Feferman’s “Enriched stratified systems for the foundations of category theory,” in What is Category Theory? [G. Sica 2006]). | |
| Mar 11, 2011 at 21:58 | comment | added | Flash Sheridan | The discussion here has also been what I call “temporalist,” e.g., “Eventually,” “new a's from old” [quoting Scott], and “already.” One comment on the temporalist metaphor and predicativity, which is not original with me (I heard it from someone at Oxford in the 80’s): If you’re serious about the metaphor, why can sets at one stage be defined via quantification over later stages? | |
| Mar 11, 2011 at 21:56 | comment | added | Flash Sheridan | “Iterative” isn’t my metaphor, so I’m not sure I’m the right person to defend it, but even Gödel, in the classic Platonist defensive of the iterative conception, uses explicitly temporal language at least once: “immediately,” [footnote 12 of “What is Cantor’s Continuum Problem,” reprinted in Benacerraf & Putnam]. Boolos uses temporal language emphatically in “The Iterative Conception of Set,” e.g., “now” twice, once italicized, on page 491 of B&P. He does of course disclaim the metaphor for the formal exposition, but it’s the motivation we’re discussing now, not the formalism. | |
| Mar 11, 2011 at 17:37 | history | edited | Flash Sheridan | CC BY-SA 2.5 |
Link to Forster’s online paper
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| Mar 11, 2011 at 14:57 | comment | added | Andreas Blass | Would a Platonist ever "believe that you can later construct new sets"? Wouldn't the most you can do "later" be to notice and describe some sets that had actually been there all along? | |
| Mar 11, 2011 at 8:30 | history | edited | Flash Sheridan | CC BY-SA 2.5 |
link Frege-Russell cardinals
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| Mar 11, 2011 at 6:23 | history | answered | Flash Sheridan | CC BY-SA 2.5 |